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2 2 2 z′ c t s . (5) | |≤ − Integral (4) for the vectorp potential A thus evaluates as ‡ √c2t2 s2 µ I − dz A(s,t)= 0 0 zˆΘ(t s/c) ′ 2 2 2 2 2 4π − √c t s √s + z′ Z− − µ I = 0 0 zˆ[ln(ct + c2t2 s2) ln s]Θ(t s/c). (6) 2π − − − The electric field is therefore given byp E(s,t)= ∂A(s,t)/∂t − µ I c = 0 0 zˆ Θ(t s/c) (7) − 2π √c2t2 s2 − − The standard convention is understood according to which f(x)Θ(x x0) = 0 whenever x x0, even‡ when the expression f(x) happens not to be defined at these values− of x. ≤ Time-dependent fields of a current-carrying wire 3 and the magnetic field by § B(s,t)= ∇ A(s,t)= (∂A /∂s)φˆ × − z µ I ct = 0 0 φˆ Θ(t s/c). (8) 2πs √c2t2 s2 − − In both (7) and (8), the delta-function terms that arose from the derivatives of the Heaviside step function in (6) dropped out on account of the property f(x)δ(x x )= f(x )δ(x x ) (9) − 0 0 − 0 of the delta function. Inspecting equations (7) and (8) we see that the fields E(s,t) and B(s,t) attain in the limit t their familiar static values 0 and (µ0I0/2πs)φˆ, respectively, and that both these→ fields ∞ diverge when t s/c. → 3. Solution using Jefimenko’s equations
As is now well known, starting from the retarded solution to the inhomogeneous wave equations for the fields E and B [4, 5], or from the familiar retarded potentials [1, 6], or otherwise [7], the time-dependent generalizations of the Coulomb and Biot-Savart laws can be derived: ˙ 1 ̺(r′,tr) ̺˙(r′,tr) J(r′,tr) 3 E(r,t)= + d r′, (10) 4πǫ 3 R c 2 RR− c2 0 Z " R R R # ˙ µ0 J(r′,tr) J(r′,tr) 3 B(r,t)= + d r′, (11) 4π 3 c 2 × R Z " R R # where ̺ is the volume charge density, r r , and the dots denote partial R ′ differentiation with respect to time. TheseR equations, ≡ − showing explicitly true sources of E and B, were first derived by Jefimenko [4]. We now shall calculate the fields E and B in the problem at hand using Jefimenko’s equations. In our problem, the charge density vanishes, ̺ =0, (12) since by assumption the wire is electrically neutral, and using equation (2) we get
I0 δ(s′) 2 2 J(s′,tr)= Θ(t s + z′ /c) zˆ, (13) 2π s′ − p I δ(s ) ˙ 0 ′ 2 2 J(s′,tr)= Θ(t s/c) δ(t s + z′ /c) zˆ. (14) 2π − s′ − p Here, cylindrical coordinates are used again, and the delta-function property (9) and the taking, with no loss of generality, the field coordinate z to be 0 reduced the retarded time tr to the same value as that in equation (4); the step-function factor in (14) expresses the fact that the step function in (13) entails that not only the
It is perhaps worthwhile to note that expression (6) for the vector potential A(s, t) resembles the §quasi-static vector potential at a distance s from the midpoint of a straight wire of finite length 2 2 2 2l = 2√c t s carrying a constant current I0. A calculation of the magnetic field according to B = ∇ A in− which the distance-dependent length 2l is treated as a constant would be equivalent to the use× of the Biot–Savart law. However, this law is not applicable beyond the quasi-static regime (cf., e.g., [3]). Time-dependent fields of a current-carrying wire 4 current density itself but also its partial time derivative vanishes for times t < s/c. Substitution into Jefimenko’s equation (10) then gives 2 2 I0 ∞ δ(s′) ∞ δ(t √s + z′ /c) E(s,t)= zˆ Θ(t s/c) s′ ds′ dz′ − , (15) 2 2 2 −4πǫ0c − 0 s′ √s + z′ Z Z−∞ which can be evaluated easily using the decomposition of the delta function [5, 8] 2 2 2 c t 2 2 2 2 2 2 δ(t s + z′ /c)= δ(z′ c t s )+ δ(z′ + c t s ) (16) − √c2t2 s2 − − − as p − h p p i µ I c E(s,t)= 0 0 zˆ Θ(t s/c), (17) − 2π √c2t2 s2 − − in full agreement with the electric field (7), obtained using the retarded vector potential. In a similar fashion, using equations (11), (13), (14) and (16) we obtain
√c2t2 s2 µ I − dz 1 ∞ δ(t √s2 + z 2/c) B 0 0 φˆ ′ ′ (s,t)= Θ(t s/c)s 2 2 3/2 + − 2 2 dz′ 4π − √c2t2 s2 (s + z′ ) c s + z′ "Z− − Z−∞ # µ I ct = 0 0 φˆ Θ(t s/c), (18) 2πs √c2t2 s2 − − in full agreement with the magnetic field (8), obtained using the retarded vector potential.
4. Discussion
At first sight, the fact that the fields E and B obtained diverge when t s/c+ while vanishing for t < s/c may seem disturbing. However, a closer examination→ reveals that, in the correct solution to the problem, E and B must tend to infinity when t s/c+. This is more transparent through the use of Jefimenko’s equations (10) and→ (11) than the use of a retarded vector potential. It is clear from equations (13), (14) and (16) that the abrupt turning of the current on at t = 0 necessarily yields an infinite time derivative of the current density, producing in the fields a cylindrical ‘shock wave’ that diverges at the time t = s/c at a distance s from the wire. Similar to the instructive example of Jackson of an abruptly turned on electric dipole [9], the diverging fields E and B are here artifacts of the unphysical, instantaneous turn-on of the current. The divergencesk disappear if the current is not turned on abruptly but is increased gradually during a short time interval τ. As a simple example, assume that the current increases linearly from zero at t = 0 to a steady non-zero value I0 at t τ, replacing accordingly expression (2) for the current density by ≥ I δ(s) t t τ J(s,t)= 0 Θ(t) − Θ(t τ) zˆ. (19) 2π s τ − τ − We remind the reader that a similar situation is found in the well-known RC-circuit problem kof charging a capacitor of capacitance C by connecting it instantaneously to a constant voltage V through a resistor of resistance R, assuming that the charge Q on the positive plate is zero at t = 0. The standard (tacit) assumption that the inductance L of the circuit is zero then leads to the equation V = Q/C + RI. In the unphysical setup of the problem (the abrupt closing of a circuit with L = 0), the correct solution must satisfy the unphysical initial condition I(t = 0) = V/R, despite the fact that the current I vanishes for t< 0. Time-dependent fields of a current-carrying wire 5
Using Jefimenko’s equations, the resulting fields are then obtained to be µ I zˆ E (s,t)= 0 0 ln ct/s + c2t2 s2/s Θ(t s/c) τ − 2π τ − − h p ln c(t τ)/s + c2(t τ)2 s2/s Θ(t τ s/c) (20) − − − − − − and p i µ I φˆ B (s,t)= 0 0 c2t2 s2Θ(t s/c) c2(t τ)2 s2Θ(t τ s/c) . (21) τ 2πc sτ − − − − − − − While the fields (20)hp and (21) are finite forp any non-zero parameter τ, theiri limits τ 0 can be shown easily to be the fields (7) and (8), respectively, that diverge when t →s/c. → There is another query. The retarded vector potential and Jefimenko’s equations are both derived under the assumption that the sources (charges and currents) are localized in a finite region of space, but our problem involves an infinitely long current- carrying wire. Therefore, the question arises as to the validity of the solution found. However, inspecting equation (6) we see that for any given finite s and t only¶ a finite segment of the wire contributes to the retarded vector potential. Figuratively speaking, retardation makes the infinitely long wire finite. Still, one should check that the vector potential (6) satisfies the requisite inhomogeneous wave equation, 1 ∂2A(s,t) µ I δ(s) 2A(s,t) = 0 0 Θ(t) zˆ, (22) ∇ − c2 ∂t2 − 2π s and that the fields (7) and (8) satisfy all Maxwell’s equations. Let us check first whether Maxwell’s equations are satisfied. Straightforward calculations yield that the fields (7) and (8) are divergenceless, confirming the equations ∇·E = ̺/ǫ0, where ̺=0, and ∇·B = 0. While a straightforward calculation of the curl of the electric field (7) confirms that the fields obtained satisfy Faraday’s law, a similar calculation of the curl of the magnetic field (8) using the standard cylindrical-coordinate formula 1 ∂(sF ) ∇ F (s)φˆ = zˆ (23) × s ∂s appears to yield only that ∇ B = ǫ0µ0∂E/∂t, instead of the full Amp`ere–Maxwell law, which reads in our case × µ I δ(s) ∂E ∇ B = 0 0 Θ(t)zˆ + ǫ µ . (24) × 2π s 0 0 ∂t Here, however, it is important to bear in mind that when the sources of a field are idealized point, line or surface distributions of charge and/or current, described by generalized functions such as the Dirac delta function, great care must be taken to employ in the field or potential differential equations generalized (distributional) derivatives instead of the usual (classical) ones+. Keeping this in mind, a careful examination of the differential operation on the magnetic field (8) implied by formula
Recall that in electrostatics the standard solution to the Poisson equation is not generally valid for charge¶ distributions extending to infinity (cf., e.g., [10, 11]). + For example, calculating the Laplacian of the potential 1/r of a unit point charge using classical derivatives yields 2(1/r) = 0 for all r > 0; at r = 0, the classical Laplacian is simply not defined. In the well-known relation∇ 2(1/r) = 4πδ(r), the Laplacian is in fact the generalized one, as it must be since this relation expresses∇ equality− of two generalized functions. To avoid confusion, some authors denote generalized differential operators by a bar, writing thus ¯ 2(1/r) = 4πδ(r) [12, 13, 14]. ∇ − Time-dependent fields of a current-carrying wire 6
(23) reveals that it involves an expression that can be written as the Laplacian of the natural logarithm of the cylindrical coordinate s,∗ 1 ∂ 1 1 ∂ ∂ s = s (ln s) = 2(ln s). (25) s ∂s s s ∂s ∂s ∇ In terms of the usual (classical) derivatives, this Laplacian vanishes for all s> 0 and is not defined at s = 0, but in view of the fact that the current density involves the delta function we should use here the relation δ(s) 2(ln s)= . (26) ∇ s Employing this relation in the evaluation of ∇ B, it is found easily that the fields (7) and (8) now satisfy the full Amp`ere–Maxwell× law (24). Since the relation (26) may appear to be novel to some readers, we give an informal proof of it using a limiting procedure in which ln s is regularized as ln √s2 + a2 and the limit a 0 is taken after integrating the product of 2 ln √s2 + a2 and a well-behaved ‘test’ function→ f(s): ∇ 2 ∞ ∞ 2a lim 2 ln s2 + a2f(s)s ds = lim f(s)s ds. (27) a 0 a 0 2 2 2 → 0 ∇ → 0 (s + a ) Z p Z The expression in square brackets is the Laplacian of ln √s2 + a2, which is now a well- behaved function of s for any a = 0; its integral over the whole plane is independent of a, equalling 2π. Splitting the6 integral into integrals over intervals 0 s S and S s< so that the function f(s) can be expanded in the first interval≤ in≤ a Taylor series≤ around∞ s = 0, the limit (27) can be evaluated as S 2 (n) n 2a ∞ f (0)s ∞ f(s) lim s ds + lim 2a2 s ds a 0 (s2 + a2)2 n! a 0 (s2 + a2)2 → Z0 n=0 → ZS X 2 f(0) S Sa 2 2 = lim + a arctg f ′(0) + O(a ,a ln a)+ = f(0). (28) a 0 1+a2/S2 a − S2+a2 ··· → The second limit in the first line vanished since the integral of f(s)s/(s2 + a2)2, where f(s) is assumed to be integrable over the whole interval (0, ), is guaranteed to converge to a value that remains finite when a 0. Having thus∞ shown that → ∞ lim 2 ln s2 + a2f(s)s ds = f(0), (29) a 0 → 0 ∇ Z p where f(s) may be any well-behaved function of s, we can write lim 2 ln s2 + a2 = δ(s)/s, (30) a 0 → ∇ p 2 2 2 2 which is the relation (26) with (ln s) lima 0 ln √s + a .♯ (Note that the relation (26) may be interpreted as∇ the Poisson≡ equation→ ∇ for the electrostatic potential (1/2πǫ0) ln s of a charge density δ(s)/2πs, which is that of an infinitely long straight −line of charge of unit line charge density [5, 10, 11, 16].) Checking whether the vector potential (6) satisfies the inhomogeneous wave equation (22) is a somewhat cumbersome but in every step straightforward calculation. ∇ B ∇ ∇· A 2A ∗ This fact is perhaps more transparent when using = ( ) and recognizing that for A expressed by equation (6), 2(ln s) appears explicitly× in the calculation−∇ of 2A. ♯ Strictly speaking, the limit here∇ and in (30) is the weak limit (cf., e.g., [8]) and∇ the Laplacian in (26) is the generalized (distributional) one (cf., e.g., [12]). Time-dependent fields of a current-carrying wire 7
Using the relation (26) and making extensive use of relation (9) in the terms containing δ(t s/c), which arise through differentiations of Θ(t s/c), we obtain − − 1 ∂2A(s,t) µ I δ(s) 2A(s,t) = 0 0 Θ(t s/c) zˆ, (31) ∇ − c2 ∂t2 − 2π s − which confirms, as it must , that the vector potential (6) satisfies the wave equation (22) since, on account of (9),†† δ(s)Θ(t s/c)= δ(s)Θ(t). Closing the discussion, we remark− that the solution to the problem of finding the fields due to an infinite straight linear wire carrying a constant current that is turned on abruptly appears to be relevant to the related problem of finding charges and fields in a current-carrying wire of finite cross section, where it appears that an infinite time is needed for establishing a stationary charge distribution [15]. However, it should be noted that the time-dependent fields obtained here approach their values at t very rapidly. For example, the field (18) at a distance s = 1 m and a time t→= ∞ 3 µs differs from its asymptotic value by less than 1 part in a million (the wire is then required to be at least 2 km long).
5. Conclusions
We calculated the electric and magnetic fields of an infinitely long wire carrying a constant current that is turned on abruptly, using Jefimenko’s equations. Our calculations confirmed the terse solution given to this problem in an example of Griffiths’s text, but our method appears to provide more insight than the standard approach via the retarded potentials: the divergence of the resulting fields E(s,t) and B(s,t) when t s/c then can be seen easily as a necessary consequence of the ‘unphysical’ setup of→ the problem. We also calculated the fields for a more realistic case in which the current in the wire is increased linearly in a nonzero time to its final steady value. We believe that our analysis was an instructive demonstration of not only the power but also the possible pitfalls of using the delta function in the calculations of electrodynamics.
Acknowledgments
We thank David Griffiths for illuminating correspondence and the anonymous referee for constructive comments. DVR acknowledges support of the Ministry of Science and Education of the Republic of Serbia, project No. 171028. VH co-authored this paper in his private capacity; no official support or endorsement by the Centers for Disease Control and Prevention is intended or should be inferred.
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